Scheduling in Multihop Wireless Networks without Back-pressure

Transcription

1 Forty-Eighth Annual Allerton Conerence Allerton House, UIUC, Illinois, USA September 29 - October 1, 2010 Scheduling in Multihop Wireless Networks without Back-pressure Shihuan Liu, Eylem Ekici, and Lei Ying Department o Electrical and Computer Engineering, Iowa State University {liush08, leiying}@iastate.edu Department o Electrical and Computer Engineering, The Ohio State University ekici@ece.osu.edu Abstract This paper ocuses on scheduling in multihop wireless networks. The well-known back-pressure scheduling algorithm is throughput optimal, but requires constant exchange o queue-length inormation among neighboring nodes or calculating the back-pressure. In this paper, we propose a selregulated MaxWeight scheduling, which does not require backpressure calculation. We prove that the sel-regulated MaxWeight scheduling is throughput optimal 1 when the traic lows are associated with ixed routes and deterministic arrivals. I. INTRODUCTION This paper considers scheduling in multihop wireless networks. A widely-used algorithm to stabilize multihop lows in wireless networks is the back-pressure algorithm proposed in [1], which can stabilize any traic lows that can be supported by any other routing/scheduling algorithm. We reer to [2], [3] or a comprehensive survey on the back-pressure algorithm and its variations. A key idea o the back-pressure algorithm is to use queue dierence as link weight, and schedule the links with large weights. Thereore, the backpressure algorithm requires constant exchange o queue-length inormation among neighboring nodes. Further, the sum o the queue-lengths along a route increases quadratically as the route length [4], which leads to poor delay perormance. The delay perormance o the back-pressure algorithm has received much attention recently. Dierent approaches have been developed to improve the delay perormance o back-pressure [4], [5], [6]. These algorithms however still rely on back-pressure or scheduling, so still require a constant exchange o queuelength inormation. In this paper, we consider the ollowing question: can the network be stabilized without using back-pressure? We address this question in a multi-hop wireless network with ixed routing. We note that a multi-hop low with a ixed route can be broken into multiple single-hop lows, one or each link on the route. A scheduling policy that stabilizes the collection o single-hop lows also provides suicient service or supporting the set o multi-hop lows. Thereore, assuming each link knows the aggregated rate it needs to carry, an alternative scheduling approach is to let each link generate virtual packets according to the aggregated rate and then let the network schedule the links according to the virtual 1 An algorithm is said to be throughput-optimal i it can stabilize any traic that can be stabilized by any other algorithm. queues. When a virtual queue is scheduled, real packets are served according to the allocated link rate. This approach is throughput optimal under the ixed routing assumption, but again requires inormation exchange in network. A source needs to estimate the arrival rate and communicate the rate to all nodes along the route associated with the source. A directly ollowing question is whether, and under what conditions, it is possible to stabilize a network without explicitly exchanging any inormation among nodes in the network. In the ollowing, we present an algorithm that can achieve this goal or networks where i sources generate packet at constant rates, and ii routes are ixed. We propose a sel-regulated MaxWeight scheduling algorithm where each node estimates the aggregated link rate locally i.e., by taking average over the past arrivals on that link. We would like to emphasize that the accuracy o linkrate estimates relies on the stability o the network because i one queue builds up, it blocks packets to down-stream nodes so that those nodes cannot accurately estimate the link rates. On the other hand, the stability o the network relies on the accuracy o the link-rate estimates, which is an interesting paradox and makes the stability o the selregulated MaxWeight scheduling a non-trivial problem. In this paper, we prove that the sel-regulated MaxWeight scheduling is throughput optimal when the traic lows are associated with ixed routes and constant arrivals. The sel-regulated MaxWeight scheduling combined with distributed scheduling algorithms such as the CSMA-based scheduling [7], [8], [9] provides a scheduling algorithm or multi-hop wireless networks, which does not require any inormation exchange in the network. Finally, we would like to comment that the proposed algorithm is motivated by the idea o regulators, which was irst proposed or re-entrant lines in [10] and later used or scheduling in wireless networks [11]. Our algorithm however does not require any inormation exchange in network, while the algorithm in [11] requires the mean arrival rates to be communicated to regulators in the network. II. BASIC MODEL In this section, we describe the basic model. We consider a network represented by a graph G N, L, where N is the set o nodes and L is the set o directed links. Denote by m, n the link rom node m to node n, which implies /10/$ IEEE 686

2 that node m can communicate with node n. Furthermore, let µ {µ m,n } denote a link-rate vector such that µ m,n is the transmission rate over link m, n. A link-rate vector µ is said to be admissible i the link-rates speciied by µ can be achieved simultaneously. Deine Γ to be the set o all admissible link-rate vectors. It is easy to see that Γ depends on the choice o intererence model and might not be a convex set. Furthermore, Γ is time-varying i channels are time-varying. Furthermore, we assume that there exists µ max such that µ m,n µ max or all m, n L and all admissible µ. We consider multihop traic lows with ixed routing in this paper. Let denote a low, and R denote the route associated with low. We use F to denote the set o all lows in the network. Assume that time is discretized, and let X F denote the number o packets injected by low at time t. In this paper, we assume X is a constant. Denote by S the source node o low, and D the destination node o low. Further, let µ m,n denote the rate at which packets o low are served over link m, n. III. NECESSARY CONDITIONS FOR STABILITY In this section, we study the necessary conditions or network stability. We say a traic coniguration {X } F is supportable i there exists { µ m,n } such that the ollowing two conditions hold: i For any low and node n D, X I {S n} + µ m,n µ n,b, 1 ii m: b:n,b L where I {S n} equals to one i S n occurs and equals to zero otherwise. F µ m,n CHΓ, 2 where CHΓ is the convex hull o Γ. Recall that each low is associated with a ixed route. It is easy to see the necessary condition 1 is equivalent to the ollowing statement: For any low and m, n R, we have µ m,n X. 3 IV. SELF-REGULATED MAXWEIGHT SCHEDULING FOR MULTIHOP WIRELESS NETWORKS In this section, we introduce the sel-regulated MaxWeight scheduling or multhop wireless networks. Two-stage queue architecture: Each node maintains two types o queues: per-low queues and per-link queues as shown in Figure 1. An incoming packet is at irst buered at the corresponding per-low queue and then moved to the per-link queue where the link is on the route o the low the details will be described later. In this paper, we denote by Q n the length o the queue maintained at node n or low, and Q n n,b the length o the queue maintained at node n or link n, b. Clearly, queue or link n, b is maintained Fig. 1. The two-stage queue architecture only at node n, so we simpliy Q n n,b to be Q n,b without causing any conusion. Sel-Regulated MaxWeight Scheduling MaxWeight scheduling: Compute the admissible link-rate vector µ t such that µ t arg max µ n,b tq n,b t. 4 µ Γ n,b L Per-link queue transmission: Node n transmits s n,b t min{µ n,b t, Q n,bt} packets to node b over link n, b. The packets are deposited into per-low queues at node b according to their lows. We let s n,bt denote the number o packets o low that are transmitted over link n, b at time t. Note that s n,b t s n,b t always holds. Per-low queue transmission: Denote by a n t the number o packets deposited into queue Q n at time slot t, i.e., a n t s b,n t. b,n L For each low, node n maintains a rate estimate t X n τ1 t 1 + γ an τ, 5 t where γ is a positive constant which can be set as small as desired. At time slot t, node n moves s n t min{ X n t, Q n t} packets rom queue Q n to queue Q n,b where b is the next hop to reach destination D rom node n in other words, b is on the route to destination D. We urther deine a n,b t s n t. :n,b R 687

3 The notations are illustrated in Figure 2. From the deinition o the sel-regulated MaxWeight scheduling algorithm, we can see that the dynamics o queue Q n and Q n,b are: Fig. 2. Q n t + 1 Qn Q n,b t + 1 The notations and the lows t sn t + an t Q n,b t s n,b t + a n,b t, In the ollowing, we will prove that the sel-regulated MaxWeight scheduling algorithm stabilizes any traic that satisies the necessary conditions. The analysis consists o two steps: we irst show that the per-link queues are bounded Lemma 1, and then using induction to prove that the per-low queues are bounded as well Lemma 2. Lemma 1: Given a set o traic lows {1 + ǫx } is supportable or some ǫ > γ, there exists a positive constant M > 0 such that the lengths o the per-link queues are no more than M or all t 0. Proo: First, note that t τ1 an τ tx according to causality, which implies that X n t 1 + γx, t. Since {1 + ǫx } is supportable, according to the necessary conditions, there exist µ such that µ m,n 1 + ǫx or all low and link m, n that is on the route o. Let µ m,n :m,n R µ m,n, then or any link m, n L, we can conclude that a m,n t Xm t Note that ǫ > γ. :m,n R 1 + γx :m,n R 1 + ǫx ǫ γx :m,n R :m,n R ǫ γ :m,n R µ m,n µ m,n ǫ γ We construct Lyapunov unction as ollows: V t Q 2 m,n t. :m,n R X :m,n R X. 6 Then V t + 1 V t t + 1 Q 2 m,n Q 2 m,n t Qm,n t Q m,n t Qm,n t + 1 Q m,n t 2Qm,n t + a m,n t s m,n t am,n t s m,n t δ + δ + + where δ Lµ max 2. 2Q m,n t a m,n t s m,n t 2Q m,n t a m,n t µ m,n 7 2Q m,n t µ m,n s m,n t, 8 From inequality 6, we can get 2Q m,n t a m,n t µ m,n ǫ γ 2Q m,n t :m,n R X. Furthermore, rom the MaxWeight scheduler 4 and necessary condition 2, 2Q m,n t µ m,n s m,n t 2δ. Thereore, we conclude that V t + 1 V t 3δ ǫ γ 2Q m,n t µ m,n µ m,n t + 2δ 2Q m,n t :m,n R X. V t Now note that max Q m,n t L. So V t > 2 2δ L ǫ γmin X implies that max Q 2δ m,nt >, ǫ γmin X 688

4 and V t + 1 V t 3δ ǫ γ 2Q m,n t 3δ 2ǫ γ max Q m,nt < δ. :m,n R X min X From the analysis above, it is easy to show that there exists a constant M V such that V t M V or all t. Since Q n,b t V t holds or all t, we can conclude the lemma by choosing M M V. Next, based on Lemma 1, we will prove that all per-low queues are bounded as well. Lemma 2: Given a set o traic lows {X } such that {1+ ǫx } is supportable or some ǫ > γ, under the sel-regulated MaxWeight scheduling, there exists a constant M such that, the lengths o the per-low queues are no more than M or all t. Proo: For the node S, the number o packets arriving at each time slot to queue Q s is a constant, i.e., a S t X, t. Since a S t is a constant, the estimate X S t is a constant as well. From 5, we have X S t 1 + γx, t. Thereore Q s t X or all t. Next we use induction to prove all per-low queues are bounded by a constant. Denote by n i the ith node on the route o low. Now assume that Q nj t M i or all t, and j i. induction assumption From Lemma 1 we know, or any traic that {1 + ǫx } is supportable and ǫ > γ, all per-link queues are bounded by a constant M. In other words, Q n i,n i+1 t M or all t. We next deine M δ im i + M. Now consider the departures o the per-low queue Q ni+1 at node n i+1. It is easy to see that tx M δ t τ1 a ni+1 τ tx because M δ is an upper bound on the number o packets belonging to low and queued at nodes n 1 to node ni the up-streaming nodes o node i + 1. We then can obtain that t τ1 ani+1 τ X ni+1 t 1 + γ, t which converges to 1 + γx as t. In other words, there exists t δ, such that when t > t δ, 1 + γx ni+1 X t > 1 + γ 2 X holds or all t. Now we deine a super time slot, each super time slot consists o M time slots, where M is a large number such that M > 2M δ γx. We index the super time slots using T. During super time slot T such that TM t δ, the number o arrivals to Q ni+1, denoted by a ni+1 T, satisies the ollowing inequality X M M δ a ni+1 T X M + M δ because otherwise, the number o packets that belong to low and are queued at the up-streaming nodes o node i+1 is more than M δ at time MT + 1 when the irst inequality ails or at time MT when the second inequality ails. According to the condition that M > 2M δ γx, we urther have that a ni+1 T 1 + γ/2x M or all T such that MT t δ. Now consider super time slot T + 1 recall that MT t δ. X ni+1 The estimates t or t such that MT + 1 t < MT + 2 satisies X ni+1 t > 1 + γ X, 2 which implies that the available service rate during super time slot T + 1 satisies X ni+1 T > 1 + γ X M a ni+1 T, 2 which implies that the available service rate or the per-low queue Q ni+1 during super time slot T + 1 is always greater than the arrivals at super time slot T. In other words, under the proposed scheme, the arrival packets at super time slot T can always be completely served at the end o super time slot T + 1. Thereore there exists an M i+1 value such that Q ni+1 t M i+1 or all, i and t. The lemma ollows rom the induction principle. Based on Lemma 1 and Lemma 2, we have the ollowing theorem. Theorem 3: Given a set o lows with constant arrival rates {X } such that {1 + ǫx } is supportable or some ǫ > γ, all queues under the sel-regulated MaxWeight algorithm are bounded. V. CONCLUSION In this paper, we considered scheduling in multihop wireless networks,and proposed the sel-regulated MaxWeight scheduling that does not require the exchange o queue-length inormation among neighboring nodes, hence completely eliminates the communication overhead when combined with recent CSMA-based scheduling algorithms. ACKNOWLEDGEMENT Research supported by NSF CNS , CNS , and CCF and the DTRA grants HDTRA and HDTRA

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